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Bounds for Coding Theory over Rings.

Niklas Gassner1, Marcus Greferath2, Joachim Rosenthal1

  • 1Institute of Mathematics, University of Zurich, 8057 Zurich, Switzerland.

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Summary
This summary is machine-generated.

This study introduces overweight, a generalized metric for coding theory over rings, and proves the Johnson bound for the homogeneous metric. These advancements extend coding theory metrics beyond finite fields.

Keywords:
Johnson boundPlotkin boundcoding theoryrings

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Area of Science:

  • Coding Theory
  • Abstract Algebra
  • Information Theory

Background:

  • Coding theory traditionally uses Hamming weight over finite fields.
  • Generalizing algebraic structures to rings necessitates new metrics beyond Hamming weight.
  • Existing metrics like Lee weight and Krotov's weight are specific cases of broader generalizations.

Purpose of the Study:

  • Introduce and define a generalized weight called 'overweight' for coding theory over rings.
  • Investigate the properties of overweight as a generalization of Lee and Krotov weights.
  • Introduce and prove the Johnson bound for the homogeneous metric in finite ring coding theory.

Main Methods:

  • Definition and exploration of the 'overweight' metric.
  • Application of established coding theory bounds (Singleton, Plotkin, sphere-packing, Gilbert-Varshamov) to the overweight.
  • Derivation of the Johnson bound for the homogeneous metric using distance sum estimates.

Main Results:

  • The 'overweight' metric is introduced, generalizing existing weights.
  • Standard coding bounds are established for the overweight.
  • The Johnson bound is proven for the homogeneous metric, filling a gap in the literature.

Conclusions:

  • The overweight and homogeneous metrics offer powerful tools for coding theory over rings.
  • The Johnson bound provides new theoretical insights for homogeneous metrics.
  • This work advances the generalization of metrics in algebraic coding theory.